Complex analysis, is the branch of mathematical analysis that concerns with analytic functions of complex variables. A complex number is separated by real and imaginary parts. For example, Z = m + in is a complex number. The dependent variable and the independent variable of any complex number may be separated into imaginary and real parts. In 1740, Euler's formula build a relationship between exponential functions and trigonometric functions called as Euler's formula.

It is defined for any real number y as e$^{iy}$ = cos y + i sin y. Euler's formula is named after mathematician Leonhard Euler. Scientist Euler is better known for his excellent and outstanding discoveries in fluid dynamics, optics, mechanics, graph theory and infinitesimal calculus. He invented various terminologies and notations such as notation pi, natural logarithm e, symbol of summation $\sum$ etc. Euler's formula not only used in mathematics but also in the fields of engineering and physics. In this section will study about Euler's formula in detail.

## Definition

Euler introduced various terms in mathematics and physics. Euler's formula provides a powerful connection between mathematics analysis and trigonometry. In other words, this formula is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function.
Euler defined fundamental relationship between complex exponential function e$^{iy}$ and the trigonometric functions sin y and cos y via the below definition:

e$^{iy}$ = Cos y + i sin y and

e$^{-iy}$ = Cos y - i sin y;

where i = Imaginary unit. Value of i = Square root of -1 i.e. $\sqrt{-1}$

e = Base of the natural logarithm

y = Real number

Cos y and Sin y = Trigonometric functions

Euler's formula provides an interpretation of the trigonometric functions, sine and cosine, as following weighted sums of the exponential function:

Sin y = Im(e$^{iy}$) = $\frac{e^{iy}-e^{-iy}}{2i}$

Cos y = Re(e$^{iy}$) = $\frac{e^{iy}+e^{-iy}}{2}$

The two equations above can be derived by adding or subtracting Euler's formulas:

e$^{iy}$ = cos y + i sin y

e$^{-iy}$ = Cos(-y) + i sin(-y) = cos y - i sin y

The formula is the point on the unit circle at angle y. This formula can be represented on a unit circle as follow:

## Proof

The Maclaurin series for exponential function:

exp(x) = e$^{x}$ = $\sum_{k=0}^{\infty}$ $\frac{x^k}{k!}$ = 1 + x + $\frac{x^2}{2!}$ + $\frac{x^3}{3!}$ + ......

Substitute x = iy in above series, we obtain

e$^{iy}$ = $\sum_{k=0}^{\infty}$ $\frac{(iy)^k}{k!}$ = 1 + (iy) + $\frac{(iy)^2}{2!}$ + $\frac{(iy)^3}{3!}$ + ......

= 1 + (iy) - $\frac{y^2}{2!}$ - $\frac{iy^3}{3!}$ + $\frac{y^4}{4!}$ + $\frac{iy^5}{5!}$ ...... (Using $i^2$ = -1, $i^3$ = -1, $i^4$ = 1, and so on)

= ( 1 - $\frac{y^2}{2!}$ + $\frac{y^4}{4!}$ - .........) + (iy - $\frac{iy^3}{3!}$ + $\frac{iy^5}{5!}$ - .........)

(Combine real and imaginary terms separately)

= ( 1 - $\frac{y^2}{2!}$ + $\frac{y^4}{4!}$ - .........) + i(y - $\frac{y^3}{3!}$ + $\frac{y^5}{5!}$ - .........) .....(1)

Since Maclaurin series for trigonometric functions, sin y and cos y are as follow:

Sin y = $\sum_{k=0}^{\infty}$ $\frac{(-1)^k y^{2k+1}}{k!}$ = y - $\frac{y^3}{3!}$ + $\frac{y^5}{5!}$ - ......

Cos y = $\sum_{k=0}^{\infty}$ $\frac{(-1)^k y^{2k}}{(2k)!}$ = 1 -  $\frac{y^2}{2!}$ + $\frac{y^4}{4!}$ - ......

Substitute in equation (1), we have

e$^{iy}$ = cos y + i sin y

Which is required result.

## Examples

Few examples on Euler's formula are given below:

Example 1: Solve e$^{ix}$ for x = $\pi$.

Solution:

We can write it as e$^{ix}$ =  cos x + i sin x

Substitute x = $\pi$

e$^{i \pi}$ =  cos ($\pi$) + i sin ($\pi$) ...(1)

cos ($\pi$) = -1

sin ($\pi$) = 0

=> e$^{i \pi}$ =  -1 + i (0)

or e$^{i \pi}$ + 1 = 0

Example 2: Find the values of $e^{i \frac{\pi}{4}}$, $e^{i \frac{\pi}{6}}$, $e^{i \frac{\pi}{2}}$ and $e^{i \frac{3\pi}{2}}$.

Solution: Using Euler's formula:

1) $e^{i \frac{\pi}{4}}$ = Cos($\frac{\pi}{4}$) + i Sin ($\frac{\pi}{4}$)

$e^{i \frac{\pi}{4}}$ = $\frac{\sqrt{2}}{2}$ + i $\frac{\sqrt{2}}{2}$ = $\frac{\sqrt{2}+i\sqrt{2}}{2}$ = $\frac{\sqrt{2}(1+i)}{2}$

=> $e^{i \frac{\pi}{4}}$ = $\frac{\sqrt{2}(1+i)}{2}$

2) $e^{i \frac{\pi}{6}}$ = Cos($\frac{\pi}{6}$) + i Sin ($\frac{\pi}{6}$)

$e^{i \frac{\pi}{6}}$ = $\frac{\sqrt{3}}{2}$ + i $\frac{1}{2}$ = $\frac{\sqrt{3}+i}{2}$

=> $e^{i \frac{\pi}{6}}$ = $\frac{\sqrt{3}+i}{2}$

3) $e^{i \frac{\pi}{2}}$ = Cos($\frac{\pi}{2}$) + i Sin ($\frac{\pi}{2}$)

$e^{i \frac{\pi}{2}}$ = 0 + i(1) = i

4) $e^{i \frac{3\pi}{2}}$ = Cos($\frac{3\pi}{2}$) + i Sin ($\frac{3\pi}{2}$)

$e^{i \frac{3\pi}{2}}$ = 0 + i(-1) = -i.